10.3 Applications to algebraic geometry and model theory

Classifying topoi bridge algebraic geometry and model theory, unifying diverse mathematical structures. They encode geometric objects like schemes and algebraic spaces as models of theories, revealing deep connections between algebra, geometry, and logic.

These topoi provide a powerful framework for studying mathematical theories and their models. By representing theories as categories of sheaves, they offer new insights into the nature of mathematical structures and their relationships.

Classifying Topoi in Algebraic Geometry and Model Theory

Schemes as classifying topoi

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• Schemes generalize algebraic varieties unify affine and projective varieties
• Locally ringed spaces glue together affine schemes (SpecA, OSpecA) where A is a commutative ring
• Geometric theory of schemes involves locally ringed spaces satisfying specific axioms (gluing, local nature)
• Zariski topology on schemes arises from prime ideals of rings reflects algebraic properties geometrically
• Classifying topos of a scheme encodes points as models of its geometric theory (Spec functor)

Algebraic spaces as classifying topoi

• Algebraic spaces generalize schemes allow for quotients by group actions not representable by schemes
• Étale topology refines Zariski topology captures local properties more effectively (formal étaleness)
• Sheaves on the étale site of a scheme define algebraic spaces (étale equivalence relations)
• Geometric theory of algebraic spaces involves étale descent conditions and local properties
• Classifying topos of an algebraic space represents models of its associated geometric theory

Existence of classifying topoi

• Coherent theories use finitary logic allow finite conjunctions, disjunctions, and existential quantification
• Geometric theories extend coherent theories include infinitary disjunctions capture more general properties
• Syntactic site construction:
  1. Form category of formulas and provable implications
  2. Define Grothendieck topology using covering families
  3. Take sheaves on this site as the classifying topos
• Universal model in classifying topos represents generic model of the theory
• Completeness theorem: every model of the theory corresponds to a point of the classifying topos
• Soundness theorem: logical consequences in the theory reflected by geometric morphisms between topoi

Classifying topoi in geometry vs model theory

• Geometric morphisms between classifying topoi correspond to interpretations between theories
• Varieties as models of geometric theories (equations, inequations in polynomial rings)
• Zariski spectra of rings form classifying topoi for theories of local rings with given global sections
• Categoricity in model theory relates to properties of classifying topoi (e.g., two-valued)
• Stability theory connects to topos-theoretic notions (e.g., coherent objects in the classifying topos)
• Stone duality generalizes to classify Boolean algebras, Heyting algebras, and certain topoi

Unifying role of classifying topoi

• Topos theory provides common language for algebraic geometry and model theory (sites, sheaves, geometric morphisms)
• Grothendieck topologies generalize classical topological spaces unify various notions of "space" (étale, fppf, crystalline)
• Categorical logic interprets mathematical theories within topoi allows for alternative foundations
• Synthetic differential geometry develops smooth infinitesimal analysis in certain topoi (well-adapted models)
• Applications extend to algebraic topology (homotopy types as ∞-topoi), functional analysis (C*-algebras as ringed topoi)